Some properties of the coupled Leonardo sequence

In this paper, we introduce a new Leonardo-type sequence, which we call the coupled Leonardo sequence. This sequence is defined by the non-homogeneous second-order recurrence $\mathcal{L}^{(k,t)}_{n+1} = k\mathcal{L}^{(k,t)}_{n} + t\mathcal{L}^{(k,t)}_{n-1} + Q(k,t)$, $n\ge 1$, with initial terms $\mathcal{L}^{(k,t)}_{0}=1$ and $\mathcal{L}^{(k,t)}_{1}=1$, where $k,\,t$ are real parameters and $Q(k,t)$ is a real-valued function. The coupling $Q(k,t)$ plays an important role and allows us to propose a new Leonardo-type generalization---originally defined in terms of the Fibonacci numbers---to the setting of an arbitrary Horadam sequence. Under suitable assumptions on $k$ and $t$, we introduce three classes of generalizations of the Leonardo sequence corresponding to the cases $k+t-1=0$, $k+t-1\neq0$, and $k=2$. We derive the Binet formula and several algebraic properties that allow us to establish some relations between the coupled Leonardo sequence and the associated Horadam sequence. The ordinary generating function is obtained, and the partial sums are examined.